Linear Operators: Spectral theory |
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Page 970
... similar integral representation for this extension . Let & be the family of all Borel subsets of R with finite measure and direct by inclusion . If x , denotes the characteristic function of the set e in & , and if ƒ is in La ( R ) ...
... similar integral representation for this extension . Let & be the family of all Borel subsets of R with finite measure and direct by inclusion . If x , denotes the characteristic function of the set e in & , and if ƒ is in La ( R ) ...
Page 1250
... similar to that of Stieltjes and differs from it by using the whole real axis ( ∞ , ∞ ) instead of [ 0 , ∞ ) . The Hausdorff moment problem is again similar but refers to a finite interval of 1250 XII . UNBOUNDED OPERATORS IN HILBERT ...
... similar to that of Stieltjes and differs from it by using the whole real axis ( ∞ , ∞ ) instead of [ 0 , ∞ ) . The Hausdorff moment problem is again similar but refers to a finite interval of 1250 XII . UNBOUNDED OPERATORS IN HILBERT ...
Page 1380
... similar . Q.E.D. 30 COROLLARY . Let τ , T , A , o ,, Ot , etc. , be as in Theorem 18 . Then an isolated point λ € Aσ ( T ) is an isolated singularity of 0 ‡ ; ( or , equivalently , of 9 ) . Moreover , pi , ( { 20 } ) is the residue at 2 ...
... similar . Q.E.D. 30 COROLLARY . Let τ , T , A , o ,, Ot , etc. , be as in Theorem 18 . Then an isolated point λ € Aσ ( T ) is an isolated singularity of 0 ‡ ; ( or , equivalently , of 9 ) . Moreover , pi , ( { 20 } ) is the residue at 2 ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary deficiency indices Definition denote dense eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero